Showing papers in "Journal of Vibration and Acoustics in 2012"

Harvesting Wind Energy Using a Galloping Piezoelectric Beam

Abstract: Galloping of structures such as transmission lines and bridges is a classical aeroelastic instability that has been considered as harmful and destructive. However, there exists potential to harness useful energy from this phenomenon. This paper focuses on harvesting wind energy that is being transferred to a galloping beam. The beam has a rigid tip body with a D-shaped cross section. Piezoelectric sheets are bonded on the top and bottom surface of the beam. During galloping, vibrational motion is input to the system due to aerodynamic forces on the D-section, which is converted into electrical energy by the piezoelectric (PZT) sheets. The relative importance of various parameters of the system such as wind speed, material properties of the beam, electrical load and beam’s natural frequency are discussed. Experimental and analytical investigations of dynamic response and power output are performed on a representative device. A maximum output power of 1.14 mW was measured at a wind velocity of 10.5 mph on a prototype device of length 235 mm and width 25 mm. A potential application for this device is to power wireless sensor networks on outdoor structures such as bridges and buildings. [DOI: 10.1115/1.4004674]

Effective Stiffening and Damping Enhancement of Structures with Strongly Nonlinear Local Attachments

Abstract: We study the stiffening and damping effects that local essentially nonlinear attachments can have on the dynamics of a primary linear structure. These local attachments can be designed to act as nonlinear energy sinks (NESs) of shock-induced energy by engaging in isolated resonance captures or resonance capture cascades with structural modes. After the introduction of the NESs, the effective stiffness and damping properties of the structure are characterized through appropriate measures, developed within this work, which are based on the energy contained within the modes of the primary structure. Three types of NESs are introduced in this work, and their effects on the stiffness and damping properties of the linear structure are studied via (local) instantaneous and (global) weightedaveraged effective stiffness and damping measures. Three different applications are considered and show that these attachments can drastically increase the effective damping properties of a two-degrees-of-freedom system and, to a lesser degree, the stiffening properties as well. An interesting finding reported herein is that the essentially nonlinear attachments can introduce significant nonlinear coupling between distinct structural modes, thus paving the way for nonlinear energy redistribution between structural modes. This feature, coupled with the well-established capacity of NESs to passively absorb and locally dissipate shock energy, can be used to create effective passive mitigation designs of structures under impulsive loads. [DOI: 10.1115/1.4005005]

Vibration Modeling of a Rigid Rotor Supported on the Lubricated Angular Contact Ball Bearings Considering Six Degrees of Freedom and Waviness on Balls and Races

Abstract: Nonlinear vibration analysis of angular contact ball bearings supporting a rigid rotor ispresented herein considering the frictional moments (load dependent and load independ-ent components of frictional moments) in the bearings. Six degrees of freedom (DOF) ofrigid rotor is considered in the dynamic modeling of the rotor-bearings system. More-over, waviness on surfaces of inner race, outer race, and ball are considered in the modelby representing it as sinusoidal functions with waviness orders of 6, 15, and 25. Twoamplitudes of waviness, 0.05 and 0.2lm, are considered in the investigation looking forthe practical aspects. The proposed model is validated with the experimental results byperforming the experiments. Moreover, the present model has also been validated withpublished results of researchers by incorporating needful changes in the DOF in the pro-posed model. Based on the computed results, it is observed that load dependent frictionalmoment (LDFM) significantly enhances the amplitudes of vibrations in comparison toload independent frictional moment (LIFM) irrespective to values of waviness amplitudeand waviness order. The influence of inner race waviness is relatively more on the vibra-tions in comparison to waviness of outer race and ball. Moreover, vibrations of systemenhance considerably at high amplitude of waviness, increase in the order of waviness,and at elevated operating parameters. [DOI: 10.1115/1.4005140]

Forced Vibrations of Supercritically Transporting Viscoelastic Beams

Abstract: This study focuses on the steady-state periodic response of supercritically transportingviscoelastic beams. In the supercritical speed range, forced vibrations are investigatedfor traveling beams via the multiscale analysis with a numerical confirmation. The forcedvibration is excited by the spatially uniform and temporally harmonic vibration of thesupporting foundation. A nonlinear integro-partial-differential equation is used to deter-mine steady responses. The straight equilibrium configuration bifurcates in multiple equi-librium positions at supercritical translating speeds. The equation is cast in the standardform of continuous gyroscopic systems via introducing a coordinate transform for nontri-vial equilibrium configuration. The natural frequencies and modes of the supercriticallytraveling beams are analyzed via the Galerkin method for the linear standard form withspace-dependent coefficients under the simply supported boundary conditions. Based onthe natural frequencies and modes, the method of multiple scales is applied to the govern-ing equation to determine steady-state responses. To confirm results via the method ofmultiple scales, a finite difference scheme is developed to calculate steady-state responsenumerically. Quantitative comparisons demonstrate that the approximate analyticalresults have rather high precision. Numerical results are also presented to show the con-tributions of foundation vibration amplitude, viscoelastic damping, and nonlinearity tothe response amplitude for the first and the second mode. [DOI: 10.1115/1.4006184]Keywords: supercritical, vibration, nonlinearity, transporting beam, Galerkin trunca-tion, multiple scales method

Vibration Properties of High-Speed Planetary Gears With Gyroscopic Effects

Abstract: This study investigates the modal property structure of high-speed planetary gears with gyroscopic effects. The vibration modes of these systems are complex-valued and speeddependent. Equally-spaced and diametrically-opposed planet spacing are considered. Three mode types exist, and these are classified as planet, rotational, and translational modes. The properties of each mode type and that these three types are the only possible types are mathematically proven. Reduced eigenvalue problems are determined for each mode type. The eigenvalues for an example high-speed planetary gear are determined over a wide range of carrier speeds. Divergence and flutter instabilities are observed at extremely high speeds. [DOI: 10.1115/1.4006646]

Harmonic Balance-Based Approach for Quasi-Periodic Motions and Stability Analysis

Abstract: Quasi-periodic motions and their stability are addressed from the point of view of different harmonic balance-based approaches. Two numerical methods are used: a generalized multidimensional version of harmonic balance and a modification of a classical solution by harmonic balance. The application to the case of a nonlinear response of a Duffing oscillator under a bi-periodic excitation has allowed a comparison of computational costs and stability evaluation results. The solutions issued from both methods are close to one another and time marching tests showing a good agreement with the harmonic balance results confirm these nonlinear responses. Besides the overall adequacy verification, the observation comparisons would underline the fact that while the 2D approach features better performance in resolution cost, the stability computation turns out to be of more interest to be conducted by the modified 1D approach.

Effective Dynamic Properties and Multi-Resonant Design of Acoustic Metamaterials

Abstract: In the study, a retrieval approach is extended to determine the effective dynamic properties of a finite multilayered acoustic metamaterial based on the theoretical reflection and transmission analysis. The accuracy of the method is verified through a comparison of wave dispersion curve predictions from the homogeneous effective medium and the exact solution. A multiresonant design is then suggested for the desirable multiple wave band gaps by using a finite acoustic metamaterial slab. Finally, the band gap behavior and kinetic energy transfer mechanism in a multilayered composite with a periodic microstructure are studied to demonstrate the difference between the Bragg scattering mechanism and the locally resonant mechanism.

Parametric Instability of Planetary Gears Having Elastic Continuum Ring Gears

Abstract: The parametric instability of planetary gears having elastic continuum ring gears is analytically investigated based on a hybrid continuous-discrete model. Mesh stiffness variations of the sun-planet and ring-planet meshes caused by the changing number of teeth in contact are the source of parametric instability. The natural frequencies of the time invariant system are either distinct or degenerate with multiplicity two, which indicates three types of combination instabilities: distinct-distinct, distinct-degenerate, and degenerate-degenerate instabilities. By using the structured modal properties of planetary gears and the method of multiple scales, the instability boundaries are obtained as simple expressions in terms of mesh parameters. Instability existence rules for in-phase and sequentially phased planet meshes are also discovered. For in-phase planet meshes, instability existence depends only on the type of gear mesh deformation. For sequentially phased planet meshes, the number of teeth on the sun (or the ring) and the type of gear mesh deformation govern the instability existence. The instability boundaries are validated numerically.

Resonances of a Forced Mathieu Equation With Reference to Wind Turbine Blades

Abstract: A horizontal axis wind turbine blade in steady rotation endures cyclic transverse loading due to wind shear, tower shadowing and gravity, and a cyclic gravitational axial loading at the same fundamental frequency. These direct and parametric excitations motivate the consideration of a forced Mathieu equation with cubic nonlinearity to model its dynamic behavior. This equation is analyzed for resonances by using the method of multiple scales. Superharmonic and subharmonic resonances occur. The effect of various parameters on the response of the system is demonstrated using the amplitude-frequency curve. Order-two superharmonic resonance persists for the linear system. While the order-two subharmonic response level is dependent on the ratio of parametric excitation and damping, nonlinearity is essential for the order-two subharmonic resonance. Order-three resonances are present in the system as well and they are similar to those of the Duffing equation.

Effects of Axial Load and Elastic Matrix on Flexural Wave Propagation in Nanotube With Nonlocal Timoshenko Beam Model

Abstract: In this paper, the effects of the axial load and the elastic matrix on the flexural wave in the carbon nanotube are studied Based on the nonlocal continuum theory and the Timoshenko beam model, the equation of the flexural wave motion is derived The dispersion relation between the frequency and the wave number is illustrated The characteristics of the flexural wave propagation in the carbon nanotube embedded in the elastic matrix with the axial load are analyzed The wave frequency and the phase velocity are presented with different wave numbers Furthermore, the small scale effects on the wave properties are discussed

Parametric Stability of Axially Accelerating Viscoelastic Beams With the Recognition of Longitudinally Varying Tensions

Abstract: In this paper, the parametric stability of axially accelerating viscoelastic beams is revisited. The effects of the longitudinally varying tension due to the axial acceleration are highlighted, while the tension was approximately assumed to be longitudinally uniform in previous studies. The dependence of the tension on the finite support rigidity is also considered. The generalized Hamilton principle and the Kelvin viscoelastic constitutive relation are applied to establish the governing equations and the associated boundary conditions for coupled planar motion of the beam. The governing equations are linearized into the governing equation in the transverse direction and the expression of the longitudinally varying tension. The method of multiple scales is employed to analyze the parametric stability of transverse motion. The stability boundaries are derived from the solvability conditions and the Routh-Hurwitz criterion for principal and sum resonances. In terms of stability boundaries, the governing equations with or without the longitudinal variance of tension are compared and the effects of the finite support rigidity are also examined. Some numerical examples are presented to demonstrate the effects of the stiffness, the viscosity, and the mean axial speed on the stability boundaries. The differential quadrature scheme is developed to numerically solve the governing equation, and the computational results confirm the outcomes of the method of multiple scales. [DOI: 10.1115/1.4004672]

Wave Analysis of In-Plane Vibrations of L-Shaped and Portal Planar Frame Structures

Abstract: This paper concerns in-plane vibration analysis of coupled bending and longitudinal vibrations in L-shaped and portal planar frame structures. An exact analytical solution is obtained using wave vibration approach. The classical Euler-Bernoulli as well as the advanced Timoshenko bending theories are applied in modeling the flexural vibrations in planar frames. Reflection and transmission matrices corresponding to incident waves arriving at the “L” joint from various directions are obtained. A concise and systematic approach to both free and forced vibration analysis of coupled bending and longitudinal vibrations in L-shaped and portal planar frame structures is presented. Results are compared to the Euler-Bernoulli model results available in the literature. Good agreements have been reached.

On Boundary Damping for an Axially Moving Tensioned Beam

Abstract: In this paper, an initial-boundary value problem for a linear-homogeneous axially moving tensioned beam equation is considered. One end of the beam is assumed to be simplysupported and to the other end of the beam a spring and a dashpot are attached, where the damping generated by the dashpot is assumed to be small. In this paper only boundary damping is considered. The problem can be used as a simple model to describe the vertical vibrations of a conveyor belt, for which the velocity is assumed to be constant and relatively small compared to the wave speed. A multiple time-scales perturbation method is used to construct formal asymptotic approximations of the solutions, and it is shown how different oscillation modes are damped. [DOI: 10.1115/1.4005025]

Exact vibration solution for exponentially tapered cantilever with tip mass

Abstract: This technical note is concerned with the free vibration problem of a cantilever beam with constant thickness and exponentially decaying width. Existing analytical results for such a vibration beam problem are found to be incomplete because lower frequencies could not be obtained. Presented herein is the exact characteristic equation for generating the complete vibration frequencies for the considered vibrating beam problem. Also the note treated for the first time such a tapered cantilever beam with a tip mass. The exact solutions (frequencies and mode shapes) are important to engineers designing such tapered beams and the results serve as benchmarks for assessing the validity, convergence and accuracy of numerical methods and solutions.

Nonlinear Vibration Analysis of the Wind Turbine Blade (Occurrence of the Superharmonic Resonance in the Out of Plane Vibration of the Elastic Blade)

Abstract: The use of wind turbine generator has rapidly spread as a one of the foremost clean energy sources. Recently, as the size of the wind turbine generator has become larger, its maintenance has become more difficult. However, there are few studies on the vibration analysis and its suppression in the conventional researches. The wind turbine is a special type of rotating machinery which has a long heavy blade rotating in the vertical plane under the action of the gravitational force. The wind power acting on the wind turbine blade varies periodically because of the height-dependent characteristics of the wind. Therefore, the dynamical design and analysis of the wind turbine blade requires a more thorough study. This paper investigates the fundamental vibration characteristic of an elastic blade of the wind turbine. The nonlinear vibration analysis of the superharmonic resonance is performed, and its characteristics are explained. Furthermore, the effect of the interaction of both the gravitational force and the wind force on the superharmonic resonance is clarified.

A Comparison of the Effects of Nonlinear Damping on the Free Vibration of a Single-Degree-of-Freedom System

Abstract: This article concerns the free vibration of a single-degree-of-freedom (SDOF) system with three types of nonlinear damping. One system considered is where the spring and the damper are connected to the mass so that they are orthogonal, and the vibration is in the direction of the spring. It is shown that, provided the displacement is small, this system behaves in a similar way to the conventional SDOF system with cubic damping, in which the spring and the damper are connected so they act in the same direction. For completeness, these systems are compared with a conventional SDOF system with quadratic damping. By transforming all the equations of motion of the systems so that the damping force is proportional to the product of a displacement dependent term and velocity, then all the systems can be directly compared. It is seen that the system with cubic damping is worse than that with quadratic damping for the attenuation of free vibration.

Failure Diagnosis of a Gear Box by Recurrences

Abstract: The recurrence analysis method is used in the mechanical diagnosis of a gear transmission system using time domain data. The recurrence is a natural behavior of a periodic motion system, which tells the state of the system, after running some time, and will approach a certain past state. In this paper, some statistical parameters of recurrence qualification analysis are extensively evaluated for the use of mechanical diagnosis, based on fairly short acceleration time series; recurrence results are compared with those obtained from Fourier analysis, and the identification procedures for the failure gear transmission by recurrences is also presented. It is found that, using only fairly short time series, some statistical parameters in quantification recurrence analysis can give clear-cut distinction between a healthy and damaged state.

A Journal Bearing With Variable Geometry for the Reduction of the Maximum Amplitude During Passage Through Resonance

Abstract: A concept for a journal bearing with variable stiffness and damping properties is developed in order to decrease the vibration amplitude of a rotor-journal bearing system during passage through resonance. The introduction of an additional fluid film thickness in the bearing is proposed in this work in order to alter the dynamic properties in the bearing. The bearing ring is divided into two parts with the upper part being fixed with the housing and the lower part being flexibly mounted by a preloaded spring in parallel with a viscous damper. This allows relative motion between the two parts of the bearing ring. The relative motion introduces an additional fluid film zone in the bearing under the passive displacement of the lower part due to increased impedance forces that are developed in the lubricant film at resonance operation. The general concept is to change the system’s damping and stiffness coefficients using this extra fluid film thickness only when the system passes through its critical speed in order to quench the vibration amplitude. For rotational speeds outside of the resonant regions, the bearing is considered to be fixed in order to behave as it was designed under the nominal loading operational conditions. [DOI: 10.1115/1.4007242]

Vibro-Acoustic Design Optimization Study to Improve the Sound Pressure Level Inside the Passenger Cabin

Abstract: The interior noise inside the passenger cabin of automobiles can be classified as structure-borne or airborne. In this study, we investigate the structure-borne noise, which is mainly caused by the vibrating panels enclosing the vehicle. Excitation coming from the engine causes the panels to vibrate at their resonance frequencies. These vibrating panels cause a change in the sound pressure level within the passenger cabin, and consequently generating an undesirable booming noise. It is critical to understand the dynamics of the vehicle, and more importantly, how it interacts with the air inside the cabin. Two methodologies were used by coupling them to predict the sound pressure level inside the passenger cabin of a commercial vehicle. The Finite Element Method (FEM) was used for the structural analysis of the vehicle, and the Boundary Element Method (BEM) was integrated with the results obtained from FEM for the acoustic analysis of the cabin. The adopted FEM-BEM approach can be utilized to predict the sound pressure level inside the passenger cabin, and also to determine the contribution of each radiating panel to the interior noise level. The design parameters of the most influential radiating panels (i.e., thickness) can then be optimized to reduce the interior noise based on the three performance metrics. A structured parametric study, based on techniques from the field of industrial design of experiments (DOE) was employed to understand the relationship between the design parameters and the performance metrics. A DOE study was performed for each metric to identify the components that have the highest contribution to the sound pressure levels inside the cabin. For each run, the vibro-acoustic analysis of the system is performed, the sound pressure levels are calculated as a function of engine speed and then the performance metrics are calculated. The highest contributors (design parameters) to each performance metric are identified and regression models are built to be used for optimization studies. Then, preliminary optimization runs are employed to improve the interior sound pressure levels by finding the optimum configurations for the panel thicknesses. Our results show that the methodology developed in this study can be effectively used for improving the design of the panels to reduce interior noise when the vibro-acoustic response is chosen as the performance criteria.

Approximate Entropy Analysis of the Acoustic Emission From Defects in Rolling Element Bearings

Abstract: This paper introduces approximate entropy (ApEn) to address a nonlinear feature parameter of acoustic emission (AE) signal for the defect detection of rolling element bearings. With respect to AE signal, parameter selection of ApEn calculation is investigated, and appropriate parameters are suggested. Finally, an experimental study is presented to investigate the influence of various running conditions, i.e., radial load, rotating speed and defect size, on ApEn calculation. The results demonstrate that ApEn provides an effective measure for AE analysis and can be used as an effective feature parameter of AE signal for the defect detection of rolling element bearings.

Rapid Heating Induced Vibration of Magnetostrictive Functionally Graded Material Plates

Abstract: The thermal vibration study of magnetostrictive functionally graded material (FGM) plate under rapid heating is computed by using the generalized differential quadrature (GDQ) method. The dynamic equilibrium differential equations with displacements and shear rotations of magnetostrictive FGM plate under the rapid heating are normalized and discretized into the dynamic discretized equations. The computational solutions of magnetostrictive FGM plate with four simply supported edges are obtained. Some parametric effects on the magnetostrictive FGM plates are analyzed, they are: thickness of mounted magnetostrictive layer, control gains of the proportional negative derivative, rapid heating flux values, and power law index values of FGM plate.

Computational Model for Investigating the Influence of Unbalanced Magnetic Pull on the Radial Vibration of Large Hydro-Turbine Generators

Abstract: A radial unbalanced magnetic pull (UMP) can be produced by an eccentric rotor and leads vibrations in large hydro-turbine generators The influence of nonlinear UMP on the radial vibration of a large hydro-turbine generator is analyzed in this paper The UMP is determined as a function of eccentricities and field currents by means of a simple analytical method instead of the finite element (FE) method The analytical method employs the no-load characteristic curve of an electrical machine and saturation effects of the ferromagnetic materials are taken into consideration FE rotor model of a large hydro-turbine generator unit, taking account of guide bearings, thrust bearing and periodic forces, is developed to investigate the influence of UMP on radial vibrations The FE rotor model and the analytical method for UMP constitute the computational model UMP is calculated under different rotor eccentricities and field currents by the proposed method Comparing with other analytical methods, the effectiveness of the proposed method is verified Dynamic responses of the FE model under different analytical methods for UMP are calculated to investigate the difference in vibration between different analytical methods A simulated excitation test is performed and a comparative analysis between the calculated results and the field data is provided The computational model is proved to be reasonable according to the analysis

Multi-Dimensional Harmonic Balance With Uncertainties Applied to Rotor Dynamics

Abstract: This paper describes the coupling of a Multi-Dimensional Harmonic Balance Method (MHBM) with a Polynomial Chaos Expansion (PCE) to determine the dynamic response of quasi-periodic dynamic systems subjected to multiple excitations and uncertainties. The proposed method will be applied to a rotor system excited at its support. Uncertainties considered include both material and geometrical parameters as well as excitation sources. To demonstrate the effectiveness and validity of the proposed numerical approach, the results that include mean, variation of the response, envelopes of the Frequency Response Functions and orbits will be systematically compared to a classical Monte Carlo approach.

Effect of Planetary Gear Carrier-Plate Cracks on Vibration Spectrum

Abstract: This paper examines the problem of identifying cracks in planetary gear systems through the use of vibration sensors on the stationary gearbox housing. In particular, the effect of unequal spacing of planet gears relative to the rotating carrier plate on various frequency components in the vibration spectra is studied. The mathematical analysis is validated with experimental data comparing the vibration signature of helicopter transmissions operating either normally or with damage, leading to shifts in the planet gear positions. The theory presented is able to explain certain features and trends in the measured vibration signals of healthy and faulty transmissions. The characterization offered may serve as a means of detecting damage in planetary gear systems.

Dynamic Analysis of Circular Cylindrical Shells With General Boundary Conditions Using Modified Fourier Series Method

Abstract: Dynamic behavior of cylindrical shell structures is an important research topic since they have been extensively used in practical engineering applications. However, the dynamic analysis of circular cylindrical shells with general boundary conditions is rarely studied in the literature probably because of a lack of viable analytical or numerical techniques. In addition, the use of existing solution procedures, which are often only customized for a specific set of different boundary conditions, can easily be inundated by the variety of possible boundary conditions encountered in practice. For instance, even only considering the classical (homogeneous) boundary conditions, one will have a total of 136 different combinations. In this investigation, the flexural and in-plane displacements are generally sought, regardless of boundary conditions, as a simple Fourier series supplemented by several closed-form functions. As a result, a unified analytical method is generally developed for the vibration analysis of circular cylindrical shells with arbitrary boundary conditions including all the classical ones. The Rayleigh-Ritz method is employed to find the displacement solutions. Several examples are given to demonstrate the accuracy and convergence of the current solutions. The modal characteristics and vibration responses of elastically supported shells are discussed for various restraining stiffnesses and configurations. Although the stiffness distributions are here considered to be uniform along the circumferences, the current method can be readily extended to cylindrical shells with nonuniform elastic restraints.

Extreme-Value-Based Statistical Bounding of Low, Mid, and High Frequency Responses of a Forced Plate With Random Boundary Conditions

Abstract: The problem of statistically bounding the response of an engineering structure with random boundary conditions is addressed across the entire frequency range: from the low, through the mid, to the high frequency region. Extreme-value-based bounding of both the FRF and the energy density response is examined for a rectangular linear plate with harmonic point forcing. The proposed extreme-value (EV) approach, previously tested only in the low frequency region for uncoupled and acoustically-coupled uncertain structures, is examined here in the mid and high frequency regions, in addition to testing at low frequencies. EV-based bounding uses an asymptotic threshold exceedance model of Type-I, to extrapolate the m-observational return period to an arbitrarily-large batch of structures. It does this by repeatedly calibrating the threshold model at discrete frequencies using a small sample of response data generated by Monte Carlo simulation or measurement. Here the discrete singular convolution (DSC) method a transfrequency computation approach for deterministic vibration - is used to generate Monte Carlo samples. The accuracy of the DSC method is first verified i) in terms of the spatial distribution of total energy density, and ii) across the frequency range, by comparison with a mode superposition method and Statistical Energy Analysis (SEA). EV-based bound extrapolations of the receptance FRF and total energy density are then compared with: i) directly-estimated bounds using a full set of Monte Carlo simulations, and ii) with total mean energy levels obtained with SEA. The paper shows that for a rectangular plate structure with random boundary conditions, EV-based statistical bounding of both the FRF and total energy density response is generally applicable across the entire frequency range.